Hello, MathOverflow community!

Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil of elliptic curves over $\mathbb{C}$. Under certain conditions, it is possible to express $f$ as the inverse function of the ratio of two linearly independent solutions of a second-order linear differential equation. The prototypical example is the case of the period integrals of the Legendre elliptic curve $y^2=x(x-1)(x-\lambda)$, which satisfy the Fuchsian equation $$\lambda(1-\lambda)D^2y + (1-2\lambda)Dy - y/4=0,$$ where $D=d/d\lambda$. We can interpret this differential equation as measuring the variation of the periods of an elliptic curve, as the parameter $\lambda$ changes.

Now my question is : have other Picard-Fuchs equations been calculated for modular functions? In principle, there should be many such equations; the Picard-Fuchs equation for Klein's $j$ function, without the calculation, is given in (Harnad, McKay). I have seen the calculation for the $\lambda$ case carried out in a few books. But I have not seen such equations for the Hauptmodul associated to the other genus $0$ modular curves.

Any thoughts, comments, questions or references are much appreciated.

Please be kind, as I am only an undergraduate. (There seems to be much "tough love" here!)

1-2-3 of Modular forms. $\endgroup$ – S. Carnahan♦ Jun 15 '10 at 6:19